Question

What two numbers multiply to get -126 and add to get -5

Answer 1

`xy=-126\\ x+y=-5\\\\ xy=-126\\ x=-5-y\\\\ (-5-y)y=-126\\ -5y-y^2+126=0\\ -y^2-5y+126=0\\-y^2-14y+9y+126=0\\ -y(y+14)+9(y+14)=0\\ -(y-9)(y+14)=0\\ y=9 \vee y=-14\\\\ x=-5-9 \vee x=-5-(-14)\\ x=-14 \vee x=9`

Those numbers are -14 and 9.

Answer 2

`x,y- unknown\ numbers\\\\\ \left \{ {{xy=-126} \atop {x+y=-5}} \right.\\\\\ \left \{ {{xy=-126} \atop {x=-5-y}} \right. \\\\\Substituting x=-5-y\ into\ the\ first\ equation:\\\\(-5-y)y=-126\\\\-5y-y^2=-126\\\\-y^2-5y+126=0 \\\\Finding\ roots:\\\\We\ are\ calculating\ delta\\\\ \Delta=b^2-4ac\\\\a=-1,\ b=-5,\ c=126 \\\\\Delta=(-5)^2-4*(-1)*126=25+504=529\\\\ y_1=(-b-√(\Delta))/(2a)=(5-23)/(-2)=(-18)/(-2)=9\\\\ y_1=(-b+√(\Delta))/(2a)=(5+23)/(-2)=(28)/(-2)=-14\\\\`
`Numbers\ are: \ 9 \ and\ -14.`